direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.D12, C4⋊C4⋊40D6, D6⋊2(C2×Q8), C4.68(C2×D12), C6⋊3(C22⋊Q8), (C22×S3)⋊5Q8, C12.221(C2×D4), (C2×C12).201D4, (C2×C4).156D12, (C2×C6).53C24, C6.10(C22×D4), C22.34(S3×Q8), C6.25(C22×Q8), D6⋊C4.93C22, C4⋊Dic3⋊53C22, (C22×C4).377D6, C2.12(C22×D12), C22.68(C2×D12), (C2×C12).488C23, (C22×Dic6)⋊14C2, (C2×Dic6)⋊60C22, C22.87(S3×C23), C23.340(C22×S3), (C22×C6).402C23, (C2×Dic3).15C23, C22.74(D4⋊2S3), (S3×C23).102C22, (C22×S3).159C23, (C22×C12).218C22, (C22×Dic3).82C22, C2.8(C2×S3×Q8), (C6×C4⋊C4)⋊15C2, (C2×C4⋊C4)⋊18S3, C3⋊3(C2×C22⋊Q8), C6.72(C2×C4○D4), (S3×C22×C4).5C2, (C2×C6).94(C2×Q8), (C3×C4⋊C4)⋊48C22, (C2×D6⋊C4).19C2, (C2×C4⋊Dic3)⋊21C2, (C2×C6).175(C2×D4), C2.15(C2×D4⋊2S3), (S3×C2×C4).244C22, (C2×C6).172(C4○D4), (C2×C4).143(C22×S3), SmallGroup(192,1068)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 792 in 322 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×6], C22 [×16], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×24], Q8 [×8], C23, C23 [×10], Dic3 [×6], C12 [×4], C12 [×4], D6 [×4], D6 [×12], C2×C6, C2×C6 [×6], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×11], C2×Q8 [×8], C24, Dic6 [×8], C4×S3 [×8], C2×Dic3 [×6], C2×Dic3 [×6], C2×C12 [×10], C2×C12 [×4], C22×S3 [×6], C22×S3 [×4], C22×C6, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, C4⋊Dic3 [×8], D6⋊C4 [×8], C3×C4⋊C4 [×4], C2×Dic6 [×4], C2×Dic6 [×4], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23, C2×C22⋊Q8, C4.D12 [×8], C2×C4⋊Dic3 [×2], C2×D6⋊C4 [×2], C6×C4⋊C4, C22×Dic6, S3×C22×C4, C2×C4.D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×D12 [×6], D4⋊2S3 [×2], S3×Q8 [×2], S3×C23, C2×C22⋊Q8, C4.D12 [×4], C22×D12, C2×D4⋊2S3, C2×S3×Q8, C2×C4.D12
Generators and relations
G = < a,b,c,d | a2=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >
►On 96 points
(1 57)(2 58)(3 59)(4 60)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 61)(24 62)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)(73 96)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)(81 92)(82 93)(83 94)(84 95)
(1 34 16 91)(2 92 17 35)(3 36 18 93)(4 94 19 25)(5 26 20 95)(6 96 21 27)(7 28 22 85)(8 86 23 29)(9 30 24 87)(10 88 13 31)(11 32 14 89)(12 90 15 33)(37 56 79 65)(38 66 80 57)(39 58 81 67)(40 68 82 59)(41 60 83 69)(42 70 84 49)(43 50 73 71)(44 72 74 51)(45 52 75 61)(46 62 76 53)(47 54 77 63)(48 64 78 55)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 56 16 65)(2 64 17 55)(3 54 18 63)(4 62 19 53)(5 52 20 61)(6 72 21 51)(7 50 22 71)(8 70 23 49)(9 60 24 69)(10 68 13 59)(11 58 14 67)(12 66 15 57)(25 76 94 46)(26 45 95 75)(27 74 96 44)(28 43 85 73)(29 84 86 42)(30 41 87 83)(31 82 88 40)(32 39 89 81)(33 80 90 38)(34 37 91 79)(35 78 92 48)(36 47 93 77)
G:=sub<Sym(96)| (1,57)(2,58)(3,59)(4,60)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,61)(24,62)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(73,96)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95), (1,34,16,91)(2,92,17,35)(3,36,18,93)(4,94,19,25)(5,26,20,95)(6,96,21,27)(7,28,22,85)(8,86,23,29)(9,30,24,87)(10,88,13,31)(11,32,14,89)(12,90,15,33)(37,56,79,65)(38,66,80,57)(39,58,81,67)(40,68,82,59)(41,60,83,69)(42,70,84,49)(43,50,73,71)(44,72,74,51)(45,52,75,61)(46,62,76,53)(47,54,77,63)(48,64,78,55), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,56,16,65)(2,64,17,55)(3,54,18,63)(4,62,19,53)(5,52,20,61)(6,72,21,51)(7,50,22,71)(8,70,23,49)(9,60,24,69)(10,68,13,59)(11,58,14,67)(12,66,15,57)(25,76,94,46)(26,45,95,75)(27,74,96,44)(28,43,85,73)(29,84,86,42)(30,41,87,83)(31,82,88,40)(32,39,89,81)(33,80,90,38)(34,37,91,79)(35,78,92,48)(36,47,93,77)>;
G:=Group( (1,57)(2,58)(3,59)(4,60)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,61)(24,62)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(73,96)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95), (1,34,16,91)(2,92,17,35)(3,36,18,93)(4,94,19,25)(5,26,20,95)(6,96,21,27)(7,28,22,85)(8,86,23,29)(9,30,24,87)(10,88,13,31)(11,32,14,89)(12,90,15,33)(37,56,79,65)(38,66,80,57)(39,58,81,67)(40,68,82,59)(41,60,83,69)(42,70,84,49)(43,50,73,71)(44,72,74,51)(45,52,75,61)(46,62,76,53)(47,54,77,63)(48,64,78,55), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,56,16,65)(2,64,17,55)(3,54,18,63)(4,62,19,53)(5,52,20,61)(6,72,21,51)(7,50,22,71)(8,70,23,49)(9,60,24,69)(10,68,13,59)(11,58,14,67)(12,66,15,57)(25,76,94,46)(26,45,95,75)(27,74,96,44)(28,43,85,73)(29,84,86,42)(30,41,87,83)(31,82,88,40)(32,39,89,81)(33,80,90,38)(34,37,91,79)(35,78,92,48)(36,47,93,77) );
G=PermutationGroup([(1,57),(2,58),(3,59),(4,60),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,61),(24,62),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40),(73,96),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,91),(81,92),(82,93),(83,94),(84,95)], [(1,34,16,91),(2,92,17,35),(3,36,18,93),(4,94,19,25),(5,26,20,95),(6,96,21,27),(7,28,22,85),(8,86,23,29),(9,30,24,87),(10,88,13,31),(11,32,14,89),(12,90,15,33),(37,56,79,65),(38,66,80,57),(39,58,81,67),(40,68,82,59),(41,60,83,69),(42,70,84,49),(43,50,73,71),(44,72,74,51),(45,52,75,61),(46,62,76,53),(47,54,77,63),(48,64,78,55)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,56,16,65),(2,64,17,55),(3,54,18,63),(4,62,19,53),(5,52,20,61),(6,72,21,51),(7,50,22,71),(8,70,23,49),(9,60,24,69),(10,68,13,59),(11,58,14,67),(12,66,15,57),(25,76,94,46),(26,45,95,75),(27,74,96,44),(28,43,85,73),(29,84,86,42),(30,41,87,83),(31,82,88,40),(32,39,89,81),(33,80,90,38),(34,37,91,79),(35,78,92,48),(36,47,93,77)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 3 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 3 | 6 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,7,3,0,0,0,0,10,10],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,7,3,0,0,0,0,10,6] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | D12 | D4⋊2S3 | S3×Q8 |
| kernel | C2×C4.D12 | C4.D12 | C2×C4⋊Dic3 | C2×D6⋊C4 | C6×C4⋊C4 | C22×Dic6 | S3×C22×C4 | C2×C4⋊C4 | C2×C12 | C22×S3 | C4⋊C4 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 3 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_4.D_{12} % in TeX
G:=Group("C2xC4.D12"); // GroupNames label
G:=SmallGroup(192,1068);
// by ID
G=gap.SmallGroup(192,1068);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations